Nilpotent conjugacy classes in the classical groups

The original title for this essay was ‘What you always wanted to know about nilpotence but were afraid to ask’. I have changed it, because that title is not likely to be an accurate description of the current version. It is my attempt to understand and explain nilpotent conjugacy classes in the classical complex semi-simple Lie algebras (and therefore also, through the exponential map, of the unipotent classes in the associated groups). There are many accounts of this material in the literature, but I try here something a bit different from what is available. For one thing, I have written much of this essay while writing programs to investigate nilpotent classes in all semi-simple complex Lie algebras. In such a task, problems arise that do not seem to be clearly dealt with in the literature. I begin with the simplest case Mn, the Lie algebra of GLn. In this case, the Jordan decomposition theorem tells us that there is a bijection of nilpotent classes with partitions of n. The main result in this case relates dominance of partitions to closures of conjugacy classes of nilpotent matrices. The original reference seems to be [Gerstenhaber:1959], which I follow loosely, although my explicit algorithmic approach seems to be new. I have also used [Collingwood-McGovern:1993], which is probably the most thorough introduction to the subject. I include for completeness, at the beginning, a discussion of partitions, particularly of Young’s raising and lowering operators (which I call shifts). In this discussion I have taken some material from [Brylawski:1972] whch although elementary does not seem to be found elsewhere. The basic question is simple to describe by an example. Every nilpotent 4× 4matrix has one of these Jordan forms: